Excel

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Chapter 12 appendix

Discrete Probability Distributions

Not all variables are continuous (or have a Normal distribution). Answers to ques­tions such as “What is the probability of 7 boys in the next 10 babies born at County Hospital?” involve discrete random variables (you cannot have a fraction of a baby). The number of male babies out of the next 10 babies is a binomial random variable. The binomial situation is satisfied here because the following conditions are met:

1. There are only two possibilities for sex: male and female (we usually say “success” and “failure”).

2. We are counting the number of “successes” in a fixed number of trials, n. Here, we are interested in the next n = 10 babies.

3. The probability of a success (or failure) is constant. The probability of a male birth is about 0.512.

4. Each trial is independent of the others. Ignoring the possibility of identical twins, babies are independent of one another.

Another type of discrete random variable typically encountered might be used in answer­ing a question such as “What is the probability that County Hospital sees at least 8 cases of flu in March, when the hospital sees 7 cases of flu in a typical winter month?” Again, we have a discrete random variable because we cannot have fractions of a flu case (a per­son either has the flu or does not). We are interested in only the number of “successes” (flu cases, not car wrecks, sprains, other illnesses, or other events that cause patients to visit the hospital) across some particular unit of time or space (here, the month of March). Additionally, we assume that two cases of flu will not present at the hospital at exactly the same instant. This type of variable has a Poisson distribution, which satisfies the following conditions:

1. We see “only” the “successes,” and are not interested in how many trials that takes.

2. No two successes can occur at exactly the same time. 3. Successes happen at some typical rate per unit of time (or space).

Probabilities for these two discrete random variables can be calculated using the discrete distribution model, or in “large” sample sizes, with a Normal approximation (see the Chapter 11 Appendix for help with Normal distribution calculations).

In all cases, we have functions that find the cumulative probability, P(X ≤ k), and P(X = k), which is sometimes called the mass. If you want to find the probability of “more than k successes” or “at least k successes,” you will need to carefully consider which numbers of successes are included in the region of interest. Then use the complement rule because P(X > k) = 1 – P(X ≤ k).

Binomial Distribution

1. Click an empty cell in the spreadsheet. 2. Formulas More Functions Statistical BINOM.DIST3. In the Number_s box, input the specified value of k (the number of successes

of interest).

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CHAPTER 12 Appendix

Sample Data Teaching Scripts (in Teaching Resources) Interactive Teaching Modules Distribution Calculator

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Sample Data Teaching Scripts (in Teaching Resources) Interactive Teaching Modules Distribution Calculator

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4. In the Trials box, input the value of n (number of trials/observations). 5. In the Probability_s box, input the value of p (probability of success). 6. If you wish to find P(X = k), input 0 or FALSE in the Cumulative box. If you

wish to find P (X ≤ k), input 1 or TRUE in the Cumulative box. 7. OK

Poisson Distribution

1. Click an empty cell in the spreadsheet. 2. Formulas ➔ More Functions ➔ Statistical ➔ POISSON.DIST 3. In the X box, input the specified value of k (the number of successes). 4. In the Mean box, input the value of μ (mean number of successes per unit of

measure). 5. If you wish to find P(X = k), input 0 or FALSE in the Cumulative box. If you

wish to find P (X ≤ k), input 1 or TRUE in the Cumulative box. 6. OK

Binomial Distribution

1. Help

2. Select Binomial for the distribution. 3. Enter p(success) and n. 4. Select the type of calculation desired (input values to obtain a probability, or

enter a probability to obtain a value). 5. Enter the direction for the probability and value(s) to look up in the lower right

section.

Note: To find P(X = k), use the q1 < X <=q2 option. Enter (k 2 1) for q1 and k for q2. For example, P(X = 3) is entered with q1 = 2 and q2 = 3. In all cases, make sure q1 is less than q2, or you will get an error message.

6. Press Enter to see shading on the distribution, and the probability.

Poisson Distribution

1. Help

2. Select Poisson for the distribution. 3. Enter the mean. 4. Select the type of calculation desired (input values to obtain a probability, or

enter a probability to obtain a value). 5. Enter the direction for the probability and value(s) to look up in the lower right

section.

Note: To find P(X = k), use the q1 < X <=q2 option. Enter (k 2 1) for q1 and k for q2. For example, P(X = 3) is entered with q1 = 2 and q2 = 3. In all cases, make sure q1 is less than q2, or you will get an error message.

6. Press Enter to see shading on the distribution, and the probability.

Note: For JMP 11 and earlier versions, the Distribution Calculator (and many other interactive simulators) can be downloaded for free from jmp.com/tools.

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Minitab

Probability Distributions Poisson ➔ ➔

Probability Distributions Binomial ➔ ➔

➔ Compute Variable

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➔ CDF.Binom in the “Function Group” at right.

➔ Compute Variable

CHAPTER 12 Appendix TA12-3

Binomial Distribution

1. Calc2. Move the radio button to Probability if you wish to find P(X = k) or

Cumulative probability if you wish to find P(X ≤ k). 3. Input the value of n (number of observations) in the Number of trials box, and

p (probability of success) in the Event probability box. 4. Because in most cases you have a specific value k in mind, move the radio

button to Input constant. In the box next to this option, enter the specified value of k.

5. OK

Poisson Distribution

1. Calc2. Move the radio button to Probability if you wish to find P(X = k) or

Cumulative probability if you wish to find P(X ≤ k). 3. Input the value of μ (mean number of successes per unit of measure) in the

Mean box. 4. Because in most cases you have a specific value k in mind, move the radio

button to Input constant. In the box next to this option, enter the specified value of k.

5. OK

In either case, to find a value given a cumulative probability—that is, to find k given P(X ≤ k)—select the Inverse cumulative probability option.

Binomial Distribution

1. Transform2. Enter a name for the new variable in Target Variable, then click in the

Numerical Expression box. 3. For P(X = k), PDF and Noncentral PDF PDF.Binom in the “Function

Group” at right. 4. For P(X ≤ k), CDF and Noncentral PDF

5. In either case, enter the parameters: k, n (number of trials), and p (success). 6. A new column will be added to the data set containing the cumulative

probability. You may have to click the Variable View tab and adjust the number of decimal places to adequately show this new variable.

Poisson Distribution

1. Transform2. Enter a name for the new variable in Target Variable, then click in the

Numerical Expression box. 3. For P(X = k), PDF and Noncentral PDF ➔ PDF.Poisson in the “Function

Group” at right. 4. For P(X ≤ k), CDF and Noncentral PDF ➔ CDF.Poisson in the “Function

Group” at right. 5. In either case, enter the parameters: k and m. 6. A new column will be added to the data set containing the cumulative

probability.

SPSS does not have an inverse function for these distributions.

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CHAPTER 12 Appendix

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Binominal Distribution

1. Distribution Calculator Binomial 2. Enter n and p. 3. To look up probabilities, select the Probability tab, then select the appropriate

symbol (e.g., =, ≤) and enter k. 4. Calculate

Poisson Distribution

1. Distribution Calculator Poisson 2. Enter lambda (the process rate). 3. To look up probabilities, select the Probability tab, then select the appropriate

symbol (e.g., =, ≤) and enter k. 4. Calculate

In either case, to find a value given a cumulative probability—that is, to find k given P(X ≤ k)—select the Quantile tab, enter P(X ≤ k), then Calculate.

TI-83/-84

Both of these options appear close to the bottom of the Distributions menu. Using the up arrow key to find them is faster than using the down arrow. The option “number” varies with the model calculator and operating system version.

Binominal Distribution

1. For P(X = k), press 2nd VARS = [DISTR] then arrow to binompdf(. Press ENTER . 2. For P(X ≤ k), press 2nd VARS = [DISTR], then arrow to binomcdf(. Press ENTER . 3. In both cases, enter n, p , and k. Enter n, then a comma, then p, then a comma,

then the value to look up. For example, you would enter binomcdf(10,0.3,2) to find P(X ≤ 2) when n = 10 and p = 0.3. ENTER4.

Poisson Distribution

1. For P(X = k), press 2nd VARS = [DISTR], then arrow to poissonpdf(. Press ENTER . 2. For P(X ≤ k), press 2nd VARS = [DISTR], then arrow to poissoncdf(. Press ENTER . 3. In both cases, enter m, k. Enter the mean, then a comma, then the value to look

up. For example, you would enter poissonpdf(3,2) to find the probability of exactly 2 events when they normally happen at a rate of 3 events per unit time/ space. ENTER4.

Binominal Distribution

1. For P(X = k), use the command dbinom(k,n,p). 2. For P(X ≤ k), use the command pbinom(k,n,p).

For example, the following command finds the probability of 10 successes out of 30 independent trials where the probability of success is 0.5.

> dbinom(10,30,.5)

[1] 0.0279816

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Poisson Distribution

1. For P(X = k), use the command dpois(k, m). 2. For P(X ≤ k), use the command ppois(k, m).

For example, the following command finds the probability of 2 or fewer events when they normally happen at a rate of 3 events per unit.

> ppois(2,3)

[1] 0.4231901

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